Discrete Mathematics with Applications, 4th Edition

Susanna Epp's DISCRETE MATHEMATICS WITH APPLICATIONS, FOURTH EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses. Cover Page 1 Half-title Page 6 Title Page 7 Copyright Page 10 Dedication Page 11 Contents 12 PREFACE 20 Themes of a Discrete Mathematics Course 20 Special Features of This Book 21 Highlights of the Fourth Edition 23 Companion Website 25 Student Solutions Manual and Study Guide 25 Organization 26 Acknowledgments 27 Chapter 1: Speaking Mathematically 29 1.1: Variables 29 1.2: The Language of Sets 34 1.3: The Language of Relations and Functions 41 Chapter 2: The Logic of Compound Statements 51 2.1: Logical Form and Logical Equivalence 51 2.2: Conditional Statements 67 2.3: Valid and Invalid Arguments 79 2.4: Application: Digital Logic Circuits 92 2.5: Application: Number Systems and Circuits for Addition 106 Chapter 3: The Logic of Quantified Statements 124 3.1: Predicates and Quantified Statements I 124 3.2: Predicates and Quantified Statements II 136 3.3: Statements with Multiple Quantifiers 145 3.4: Arguments with Quantified Statements 160 Chapter 4: Elementary Number Theory and Methods of Proof 173 4.1: Direct Proof and Counterexample I: Introduction 174 4.2: Direct Proof and Counterexample II: Rational Numbers 191 4.3: Direct Proof and Counterexample III: Divisibility 198 4.4: Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem 208 4.5: Direct Proof and Counterexample V: Floor and Ceiling 219 4.6: Indirect Argument: Contradiction and Contraposition 226 4.7: Indirect Argument: Two Classical Theorems 235 4.8: Application: Algorithms 242 Chapter 5: Sequences, Mathematical Induction, and Recursion 255 5.1: Sequences 255 5.2: Mathematical Induction I 272 5.3: Mathematical Induction II 286 5.4: Strong Mathematical Induction and the Well-Ordering Principle for the Integers 296 5.5: Application: Correctness of Algorithms 307 5.6: Defining Sequences Recursively 318 5.7: Solving Recurrence Relations by Iteration 332 5.8: Second-Order Linear Homogenous Recurrence Relations with Constant Coefficients 345 5.9: General Recursive Definitions and Structural Induction 356 Chapter 6: Set Theory 364 6.1: Set Theory: Definitions and the Element Method of Proof 364 6.2: Properties of Sets 380 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras 395 6.4: Boolean Algebras, Russell’s Paradox, and the Halting Problem 402 Chapter 7: Functions 411 7.1: Functions Defined on General Sets 411 7.2: One-to-One and Onto, Inverse Functions 425 7.3: Composition of Functions 444 7.4: Cardinality with Applications to Computability 456 Chapter 8: Relations 470 8.1: Relations on Sets 470 8.2: Reflexivity, Symmetry, and Transitivity 477 8.3: Equivalence Relations 487 8.4: Modular Arithmetic with Applications to Cryptography 506 8.5: Partial Order Relations 526 Chapter 9: Counting and Probability 544 9.1: Introduction 545 9.2: Possibility Trees and the Multiplication Rule 553 9.3: Counting Elements of Disjoint Sets: The Addition Rule 568 9.4: The Pigeonhole Principle 582 9.5: Counting Subsets of a Set: Combinations 593 9.6: r-Combinations with Repetition Allowed 612 9.7: Pascal’s Formula and the Binomial Theorem 620 9.8: Probability Axioms and Expected Value 633 9.9: Conditional Probability, Bayes’ Formula, and Independent Events 639 Chapter 10: Graphs and Trees 653 10.1: Graphs: Definitions and Basic Properties 653 10.2: Trails, Paths, and Circuits 670 10.3: Matrix Representations of Graphs 689 10.4: Isomorphisms of Graphs 703 10.5: Trees 711 10.6: Rooted Trees 722 10.7: Spanning Trees and Shortest Paths 729 Chapter 11: Analysis of Algorithm Efficiency 745 11.1: Real-Valued Functions of a Real Variable and Their Graphs 745 11.2: O-, Ω-, and Θ-Notations 753 11.3: Application: Analysis of Algorithm Efficiency I 767 11.4: Exponential and Logarithmic Functions:Graphs and Order 779 11.5: Application: Analysis of Algorithm Efficiency II 792 Chapter 12: Regular Expressions and Finite-State Automata 807 12.1: Formal Languages and Regular Expressions 808 12.2: Finite-State Automata 819 12.3: Simplifying Finite-State Automata 836 Appendix A: Properties of the Real Numbers 849 Appendix B: Solutions and Hints to Selected Exercises 852 Index 971 "Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses."--Publisher Presents an introduction to discrete mathematics. This title explains complex, abstract concepts. It also presents the major themes of discrete mathematics, and the reasoning that underlies mathematical thought.